# The Gas Laws PV = nRT.

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The Gas Laws PV = nRT

Kinetic Molecular Theory (KMT)
Particles of matter are always in motion Helps to explain differences between the 3 physical state Helps to explain properties of the 3 physical states

Solids Definite shape and volume
Particles packed together in fixed positions Strong attraction between particles Very little kinetic energy Can’t be compressed

Liquids Definite volume - meniscus indicates volume
Takes some shape of container Particles close but can move; less attraction between particles More kinetic energy than solid Can’t be compressed

Gases - Part 1 Ideal - follows all KMT assumptions
No definite shape - takes shape of container; fills container completely No definite volume - takes container’s complete volume No attraction between particles Most kinetic energy - in constant motion - but KE depends on temperature Collide between gas molecules creates pressure. More collisions = more pressure Collisions are elastic – gas molecules “bounce” off each other and do not react

Ideal Gases - Part 2 Gases can be compressed. Small number of particles within large volume of space Can diffuse - particles spread out to fill space Can effuse - pass through small openings Low density – small mass/large volume (which is why most float)

Equivalent Gas Pressures
1 atm (atmosphere) - force of atmosphere pressing down at sea level and 0ºC 760 mm Hg - height of Hg column at sea level and 0ºC (or in Hg) 760 torr - same as mm Hg 14.7 psi (pounds/in2) 1.013 bars (or 1013 mbars) kPa (or Pa) - force/area

Measuring Gas Pressures - Part 1
As air pushes downward, its force (pressure) pushes Hg up into the tube. Measuring the height of the Hg column in the tube measures the air pressure in mm Hg or torr

Measuring Gas Pressures - Part 2
(A) Absolute pressure (how a barometer works; air is the “gas”) (B) Patm > Pgas because the Hg is higher on gas (weaker) side (C) Patm < Pgas because the Hg is higher on air (weaker) side

The ABCD Gas Laws A = Avogadro’s Law
2 gases at the same temperature and pressure have equal volumes and equal number of molecules Molar volume: 22.4 L of any gas at STP has 6.02 x 1023 molecules

The ABCD Gas Laws B =Boyle’s Law
Relationship between gas volume (V) and pressure (P) when temperature (T) and moles of gas (n) are constant Equation: P1V1 = P2V2 Or to solve for V2, use

Boyle’s Law As the piston pushes downward, pressure increases from P1 to P2. Volume decreases from V1 to V2 So when P increases, V decreases. This is an INVERSE (or INDIRECT) relationship

Boyle’s Law Graph P and V High Volume P and V Low Low
Pressure (mm Hg) High

Boyle’s Law Real Life Applications
Drinking from a straw Lung Ventilation (Inhale/Exhale) Spray cans

The ABCD Gas Laws C = Charles’ Law
Relationship between gas volume (V) and temperature (T) when pressure (P) and moles of gas (n) are constant V1 = V2 or T1 T2 Temperatures must be in ºK (ºC = ºK)

Charles’ Law As temperature increases, the molecules gain energy, move faster and spread out, so volume increases Since both temperature and volume are changing in the same direction it is a DIRECT relationship. Pressure remains constant (10 N) Number of particles remains constant

Charles’ Law Graph T and V T and V High Volume Low Low
Temperature (ºK) High

Charles’ Law Real Life Application
Hot air balloons - hot air rises because volume goes up with temperature. The hot air is less dense (same mass but more volume) and so it rises.

The ABCD Gas Laws D = Dalton’s Law
In a mixture, every gas exerts its own pressure called its PARTIAL PRESSURE The total pressure in the atmosphere (or container) is the sum of all the partial pressures Ptotal = P1 + P2 +P3 etc. Dalton also proved atoms existed

Dalton’s Law of Partial Pressures

Gay-Lussac’s Law Relationship between gas temperature (T) and pressure (P) when moles of gas (n) and volume (V) are constant P1 = P2 or P1T2 = P2T1 T T2 Temperatures must be in ºK (ºC = ºK)

Gay-Lussac’s Law Graph Looks a lot like the graph for Charles’ law
T P High Pressure Direct relationship Low T P  Low High Temperature (K)

KABOOM!! Gay-Lussac’s law Real Life Applications
Inner tube for tires. Gas can’t escape so volume is constant…unless the pressure gets too high and then it… KABOOM!! Gas confined in compressed gas tank

Combined Gas Law Combines both Boyles and Charles Laws
More realistic - gas pressure and temperature can both be changing and affecting volume Temperatures must be in °K Which – temperature or pressure – affects volume most? Depends on which undergoes greatest change

Combined Gas Law Equation
P1V1 = P2V2 T T2 or P1V1T2 = P2V2T1 To solve for V2 use:

Combined Gas Law – Example Problem
A weather balloon containing helium with a volume of L rises in the atmosphere and is cooled from 27 ºC to –27 °C. The pressure on the gas is reduced from kPa to 25.0 kPa. What is the volume of the gas at the lower temperature and pressure? V2 = ? P2 = 25.0 kPa T2 = -27 °C V1 = L P1 = kPa T1 = 27 °C + 273 = 300 K + 273 = 246 K

P1V1 = P2V2 T T2 P1V1T2 = P2V2T1 110.0 kPa x L x 246 K = 25.0 kPa x V2 x 300 K 110.0 kPa x L x 246 K = kPa x V2 x 300 K 25.0 kPa x 300 K kPa x 300 K L = V2 1480 L (3 s.d.)

Ideal Gas Law Combines the ABC laws (Avogadro’s, Boyle’s, and Charles)
Not only temperature, pressure, and volume change, but also moles Can be used to determine gas density, mass, and molar mass

Ideal Gas Law Equation PV = nRT P = Pressure at standard pressure
V = Volume at STP n = moles at STP T = Temperature at standard temperature So what’s R?

R – The Gas Constant PV = nRT PV 1 atm x 22.4 L nT 0.0821 atm•L = R
Notice how R contains all the units for the variables. R’s value will only change if the pressure units change P = Pressure at standard pressure (1 atm) V = Volume at STP (22.4 L) n = moles at STP (1 mole) T = Temperature at standard temperature (273 K) PV nT 1 atm x 22.4 L 1 mole x 273 K atm•L mole•K = R = R

R – The Gas Constant other values
If pressure is measured in mm Hg (or torrs): If Pressure is measured in kPa: 62.4 mmHg•L mole•K 8.31 kPa•L mole•K If Pressure is measured in mbar: If Pressure is measured in atm: 83.1 mbar•L mole•K atm•L mole•K

Ideal gas law – example problem
A 500. g block of dry ice [CO2 (s)] becomes a gas at room temperature. What volume will the dry ice have at room temperature (25°C) and 975 kPa? PV = nRT P = 975 kPa n = 500.g/molar mass CO2 V = ? R = use kPa version T = 25°C (change to °K)

PV = nRT P = 975 kPa n = molesCO2 V = ? R = 8.31 kPa•L/mole•°K T = 298°K 975 kPa•V = ( mole)(8.31 kPa•L/mole•K)(298°K) 975 kPa•V = ( mole)(8.31 kPa•L/mole•°K)(298°K) 975 kPa kPa V = L V= 28.9 L (3 s.d.)

Graham’s Law of Effusion Diffusion of Gases
Gases effuse – pass through small openings Gases diffuse – spread out from areas of high concentration to low concentration Diffusion rate depends on kinetic energy and molar mass

Graham’s Law – Part 2 Two gases at same temperature have same average kinetic energy, therefore… Speed of diffusion and effusion depends on molar mass Heavy gases are slow, light gases are fast (inverse relationship) Velocityfast Velocityslow =

Oxygen has the highest molar mass, so it has the slowest speed
Oxygen has the highest molar mass, so it has the slowest speed. Hydrogen has the smallest molar mass; it is the fastest.

Graham’s Law – Example Problem #1
Determine the ratio of velocities for H2O and CO2 at the same temperature. Determine the molar masses and which gas is fastest. Molar Mass H20 2 H = g 1 O = 16.0 g 18.0 g/mole Molar Mass CO2 1 C = g 2 O = g 44.0 g/mole Light and Fast Slow and Heavy

Velocityfast Velocityslow = VelocityH2O VelocityCO2 = = = 1.56 So H2O is 1.56 x faster than CO2 So if H2O is 1.56 x faster than CO2 – what is the CO2’s velocity if the H2O has a velocity of 6.04 m/sec?

VelocityH2O VelocityCO2
= 1.56 6.04 m/sec VelocityCO2 1.56 = 6.04 m/sec 1.56 = Velocity CO2 3.87 m/sec = Velocity CO2